On the Proximal Point Algorithm of Hybrid-Type in Flat Hadamard Spaces with Applications
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Authors: L. Y. HARUNA, G. C. UGWUNNADI AND B. ALI
DOI: 10.46793/KgJMat2406.845H
Abstract:
In this paper, we introduce a hybrid-type proximal point algorithm for approximating zero of monotone operator in Hadamard-type spaces. We then prove that a sequence generated by the algorithm involving Mann-type iteration converges strongly to a zero of the said operator in the setting of flat Hadamard spaces. To the best of our knowledge, this result presents the first hybrid-type proximal point algorithm in the space. The result is applied to convex minimization and fixed point problems.
Keywords:
Fixed Points, monotone operators, proximal point algorithm, hybrid-type algorithm, flat Hadamard spaces.
References:
[1] B. Ali and L. Y. Haruna, Approximation methods for fixed point of further 2-generalized hybrid mappings in flat Hadamard spaces, Appl. Set-Valued Anal. Optim. 3(1) (2021), 27–37. https://doi.org/10.23952/asvao.3.2021.1.04
[2] K. O. Aremu, C. Izuchukwu, G. C. Ugwunnadi and O. T. Mewomo, On the proximal point algorithm and demimetric mappings in CAT(0) spaces, Demonstr. Math. 51 (2018), 277–294. https://doi.org/10.1515/dema-2018-0022
[3] M. Bacak, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter, Berlin, München, Boston, 2014.
[4] I. D. Berg and I. G. Nicolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata 133 (2008), 195–218. https://10.1007/s10711-008-9243-3
[5] P. Cholamjiak, The modified proximal point algorithm in CAT(0) spaces, Optim. Lett. 9 (2015), 1401–1410. https://doi.org/10.1007/s11590-014-0841-8
[6] S. Dhompongsa, B. Panyanak, On Δ-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008), 2572–2579. https://doi.org/10.1016/j.camwa.2008.05.036
[7] C. Izuchukwu, A. A. Mebawondu, K. O. Aremu, H. Abass and O. T. Mewomo, Viscosity iterative techniques for approximating a common zero of monotone operators in Hadamard spaces, Rend. Circ. Mat. Palermo (2) 69(2) (2020), 1–21. https.//doi:i0.1007/s12215-019-00415-2
[8] B. A. Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc. 141(3) (2013), 1029–1039. https://www.jstor.org/stable/23558440
[9] B. A. Kakavandi and M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal. 73 (2010), 3450–3455. https://doi.org/10.1016/j.na.2010.07.033
[10] M. A. A. Khan and P. Cholamjiak, A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces, J. Fixed Point Theory Appl. 22 (2020), Article ID 62. https//doi.org/10.1007/s11784-020-00796-3
[11] H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces, J. Aust. Math Soc. 103(1) (2017), 70–90. https://doi.org/10.1017/S1446788716000446
[12] S. Kesornprom and P. Cholamjiak, Proximal type algorithms involving linesearch and inertial techniques for split variational inclusion in Hilbert spaces with applications, Optimization 68 (2019), 2365–2391. https://doi.org/10.1080/02331934.2019.1638389
[13] W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), 3689–3696. https://doi.org/10.1016/j.na.2007.04.011
[14] K. Kunrada, N. Pholasa and P. Cholamjiak, On convergence and complexity of the modified forward-backward method involving new linesearchs for convex minimization, Math. Methods Appl. Sci. 42(5) (2019), 1352–1362. https://doi.org/10.1002/mma.5420
[15] B. Martinet, Regularisation d’inequations variationelles par approximations, Rev Francaise d’Inform et de Rech Re 3 (1970), 154–158.
[16] M. Movahedi, D. Behmardi and M. Soleimani-Damaneh, On subdifferential in Hadamard spaces, Bull. Iranian Math. Soc. 42(3) (2016), 707–717. https://www.sid.ir/en/journal/ViewPaper.aspx?id=526443
[17] G. N. Ogwo, C. Izuchukwu, K. O. Aremu and O. T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in a Hadamard space, Bull. Belg. Math. Soc. Simon Stevin 27(1) (2020), 127–152. https://doi.org/10.36045/bbms/1590199308
[18] S. Ranjbar, W-convergence of the proximal point algorithm in complete CAT(0) metric spaces, Bull. Iranian Math. Soc. 43(3) (2017), 817–834. http://bims.iranjournals.ir/article_971.html
[19] S. Ranjbar and H. Khatibzadeh, Strong and Δ-convergence to a zero of a monotone operator in CAT(0) spaces, Mediterr. J. Math. 14 (2017), Article ID 56. https;//doi.org/10.1007/s00009-017-0885-y
[20] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14(5) (1976), 877–898. https://doi.org/10.1137/0314056
[21] M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. 87 (2000), 189–202. https://doi.org/10.1007/s101079900113
[22] R. Suparatulatorn, P. Cholamjiak and S. Suantai, On solving the minimization and the fixed point problem for nonexpansive mapping in CAT(0) spaces, Optim. Methods Softw. 32 (2017), 182–192. https://doi.org/10.1080/10556788.2016.1219908
[23] M. Tahernia, S. Moradi and S. Jafari, The strong convergence of a proximal point algorithm in complete CAT(0) metric spaces, Hacet. J. Math. Stat. 49(1) (2020), 399–408. http://dx.doi.org/10.15672/hujms.470975
[24] G. C. Ugwunnadi, On the strong convergence of a modified Halpern algorithm in a CAT(0) spaces, J. Anal. 28 (2020), 471–488. https://doi.org/10.1007/s41478-019-00183-3
[25] G. Z. Eskandani and M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algorithms 80 (2019), 1155–1179. https://doi.org/10.1007/s11075-018-0521-3
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